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Properties

Label 2-326700-1.1-c1-0-116
Degree 22
Conductor 326700326700
Sign −1-1
Analytic cond. 2608.712608.71
Root an. cond. 51.075551.0755
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·7-s + 7·13-s + 7·19-s − 4·31-s − 37-s − 5·43-s + 18·49-s + 61-s + 5·67-s + 10·73-s + 4·79-s − 35·91-s + 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.88·7-s + 1.94·13-s + 1.60·19-s − 0.718·31-s − 0.164·37-s − 0.762·43-s + 18/7·49-s + 0.128·61-s + 0.610·67-s + 1.17·73-s + 0.450·79-s − 3.66·91-s + 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

Λ(s)=(326700s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 326700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(326700s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 326700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 326700326700    =    22⋅33⋅52⋅1122^{2} \cdot 3^{3} \cdot 5^{2} \cdot 11^{2}
Sign: −1-1
Analytic conductor: 2608.712608.71
Root analytic conductor: 51.075551.0755
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 326700, ( :1/2), −1)(2,\ 326700,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
5 1 1
11 1 1
good7 1+5T+pT2 1 + 5 T + p T^{2}
13 1−7T+pT2 1 - 7 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1+T+pT2 1 + T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+5T+pT2 1 + 5 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+pT2 1 + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−T+pT2 1 - T + p T^{2}
67 1−5T+pT2 1 - 5 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−5T+pT2 1 - 5 T + p T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−12.95412719965603, −12.41050002287465, −11.99770362613099, −11.43094508274840, −11.05916527815333, −10.48599276647634, −10.10276377992143, −9.541542344599910, −9.306395003570393, −8.777362321760363, −8.355876872695148, −7.684691858577129, −7.235941102439814, −6.626867942176225, −6.403849211611591, −5.841298313359651, −5.476974930267060, −4.879993734468873, −3.990029358971533, −3.540142549799227, −3.416949100832096, −2.805126335922175, −2.084579396299092, −1.247185356843009, −0.8043841076366568, 0, 0.8043841076366568, 1.247185356843009, 2.084579396299092, 2.805126335922175, 3.416949100832096, 3.540142549799227, 3.990029358971533, 4.879993734468873, 5.476974930267060, 5.841298313359651, 6.403849211611591, 6.626867942176225, 7.235941102439814, 7.684691858577129, 8.355876872695148, 8.777362321760363, 9.306395003570393, 9.541542344599910, 10.10276377992143, 10.48599276647634, 11.05916527815333, 11.43094508274840, 11.99770362613099, 12.41050002287465, 12.95412719965603

Graph of the ZZ-function along the critical line